Hello, everyone!

I would like to calculate the time complexity of a Quantum Neural Network (QNN). For this, I consider the implementation of a QNN as a series of matrix multiplications.

Suppose we have matrix **A** (size a x b) and matrix **B** (size b x c). The time complexity of multiplying **A** and **B** is *O*(abc).

Referenced link: Matrix multiplication algorithm time complexity

Now, consider a 4-qubit QNN as shown below. The input state is a 2^q-dimensional vector (q=4), which is multiplied by a series of 2^q x2^q matrices (there are 11 such matrices, as the depth of the QNN is 11).

As a result, the time complexity is *O*(11 x 2^q x 2^q) = *O*(11 x 4q) = *O*(4^q), where q=4.

Does this derivation of time complexity seem correct?

I’d be happy to discuss this further and welcome any feedback!

```
Name: PennyLane
Version: 0.34.0
- default.gaussian (PennyLane-0.34.0)
- default.mixed (PennyLane-0.34.0)
- default.qubit (PennyLane-0.34.0)
...
```